If by Download speed you mean advertised or speedtest download number and by transfer rate you mean speed you get when downloading a file, then yes, generally, all else remaining the same, if one increases the other should as well. File download rate(kBps or kilobytes ps) has a theoretical max at 1/8 th of Download speed( Kbps or kilobits ps).
There could be ofcourse other factors at play. For instance did you verify the 2Mbps Number at speedtest.net? Is the new transfer speed tested on the same server as earlier?
There are a few other posts which seem to be about similar issues, which might have solutions that could help -
What downloading speed i should expect?
Download Speed is consistently at 16kbps
In the past I have had the same effect with my ISP (in a rural area too, though this is but a coincidence).
The root of the matter was that he implemented a bandwidth limitation on port 80 only. This was especially obvious with speedtest.net, where the initial speed would peak, then decrease to less than half of its peak value.
I discovered by pure chance that this did not occur on OpenVPN, where I managed to obtain the peak speed value during the whole speedtest.net test. This was made possible by the fact that the site I connected to (my work site) has a very nice, very fast, large-bandwidth connection.
Alerted by this, I tried a large file transfer via scp, and, lo and behold, I achieved the same large speeds as with the OpenVPN, rather than the lower http speeds.
You may try the same, and see whether a bandwidth limit is applied on ports 22 (scp) and 21 (ftp). It is most illuminating to use files already significantly compressed like pdfs, since this will rule out the incidence of compression per se.
Though admittedly sloppy, these bandwidth limitations are still effective, since most people use the Internet only for downloading Web pages.
EDIT:
Strictly speaking, there is a way to test this: if you control the VPN server, you may stop any other activity on the server on port 80, and start listening for VPN connections on port 80; you will also have to modify the port you connect to on the client's program. Now, if your ISP is limiting bandwidth use on port 80, the VPN should clock at exactly the same speed as the non-VPN connection.
Best Answer
Are you sure you got control of BITS and BYTES?
When testing your speed with speedtests online, you're measuring BITS per second. When downloading files, your speed is in BYTES per second. As there is 8 bits in a byte - Your download speed "should" equal your speedtest-result /8.
Getting 8mbps on speedtest, would give you 1MB/s download-speed.
Also check your sources. (Local-FTP???) You should also run different speedtests to verify a result. You will almost never get a 100% correct answer by running a speedtest, but it will give you a pointer.
To answer the question - there is software to monitor your computers bandwidth use, but it's not likely that's the culprit.
My guess is mixed terminology if you say the speed is 5 times higher on speedtest.
Addition: I just did a speedtest on speedtest.net. In my example, I "should" get a download speed of 32,5MB/s. Now - with that speed there are other things limiting downloads (like disk-speed+++) but it should give me a pointer of what I could expect with hardware that supports it.